Otten, Gilbertson, Males, & Clark (2014)

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The Mathematical Nature of Reasoning-and-Proving Opportunities in Geometry Textbooks

Abstract

International calls have been made for reasoning-and-proving to permeate school mathematics. It is important that efforts to heed this call are grounded in an understanding of the opportunities to reason-and-prove that already exist, especially in secondary-level geometry where reasoning-and-proving opportunities are prevalent but not thoroughly studied. This analysis of six secondary-level geometry textbooks, like studies of other textbooks, characterizes the justifications given in the exposition and the reasoning-and-proving activities expected of students in the exercises. Furthermore, this study considers whether the mathematical statements included in the reasoning-and-proving opportunities are general or particular in nature. Findings include the fact that the majority of expository mathematical statements were general, whereas reasoning-and-proving exercises tended to involve particular mathematical statements. Although reasoning-and-proving opportunities were relatively numerous, it remained rare for the reasoning-and-proving process itself to be an explicit object of reflection. Relationships between these findings and the necessity principle of pedagogy are discussed.

Outline of Headings

  • Background
    • Research on Students' Reasoning-and-Proving
    • Theoretical Perspective
  • Method
    • Sample
    • Analytic Framework
      • Characterizing Mathematical Statements
      • Separating Mathematical Statements from Justification
      • Separating Opportunities to Prove from Opportunities to Explain
      • Codes Inherited from Thompson, Senk, and Johnson
      • Codes Added Based on Pilot Analysis of Geometry Textbooks
      • Statements or Exercises About Reasoning-and-Proving
    • Analytic Procedures
  • Results
    • Reasoning-and-Proving in Textbook Exposition
      • Types of Statements in Textbook Exposition
      • Types of Justifications in Textbook Exposition
      • Summary
    • Reasoning-and-Proving in Student Exercises
      • Types of Statements in Student Exercises
      • Types of Reasoning-and-Proving Activities in Student Exercises
      • Statement-Types of Proof-Focused Exercises
      • Summary
    • Comparing Textbook Exposition to Student Exercises
  • Discussion

Also

APA
Otten, S., Gilbertson, N. J., Males, L. M., & Clark, D. L. (2014). The mathematical nature of reasoning-and-proving opportunities in geometry textbooks. Mathematical Thinking and Learning, 16(1), 51–79. doi:10.1080/10986065.2014.857802
BibTeX
@article{Otten2014,
abstract = {International calls have been made for reasoning-and-proving to permeate school mathematics. It is important that efforts to heed this call are grounded in an understanding of the opportunities to reason-and-prove that already exist, especially in secondary-level geometry where reasoning-and-proving opportunities are prevalent but not thoroughly studied. This analysis of six secondary-level geometry textbooks, like studies of other textbooks, characterizes the justifications given in the exposition and the reasoning-and-proving activities expected of students in the exercises. Furthermore, this study considers whether the mathematical statements included in the reasoning-and-proving opportunities are general or particular in nature. Findings include the fact that the majority of expository mathematical statements were general, whereas reasoning-and-proving exercises tended to involve particular mathematical statements. Although reasoning-and-proving opportunities were relatively numerous, it remained rare for the reasoning-and-proving process itself to be an explicit object of reflection. Relationships between these findings and the necessity principle of pedagogy are discussed.},
author = {Otten, Samuel and Gilbertson, Nicholas J. and Males, Lorraine M. and Clark, D. Lee},
doi = {10.1080/10986065.2014.857802},
journal = {Mathematical Thinking and Learning},
number = {1},
pages = {51--79},
title = {{The mathematical nature of reasoning-and-proving opportunities in geometry textbooks}},
url = {http://www.tandfonline.com/doi/abs/10.1080/10986065.2014.857802},
volume = {16},
year = {2014}
}